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%I #5 May 21 2022 14:50:29
%S 1,1,1,1,2,1,4,1,6,1,9,1,1,12,2,1,16,5,1,20,9,1,25,16,1,30,25,1,36,39,
%T 1,1,42,56,2,1,49,80,5,1,56,109,10,1,64,147,19,1,72,192,32,1,81,249,
%U 54,1,90,315,84,1,100,396,129,1,1,110,489,190,2,1,121,600,275,5
%N Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k excedances (parts above the diagonal), zeros omitted.
%e Triangle begins:
%e 1
%e 1 1
%e 1 2
%e 1 4
%e 1 6
%e 1 9 1
%e 1 12 2
%e 1 16 5
%e 1 20 9
%e 1 25 16
%e 1 30 25
%e 1 36 39 1
%e 1 42 56 2
%e 1 49 80 5
%e 1 56 109 10
%e For example, row n = 7 counts the following partitions:
%e (1111111) (7) (43)
%e (52) (331)
%e (61)
%e (322)
%e (421)
%e (511)
%e (2221)
%e (3211)
%e (4111)
%e (22111)
%e (31111)
%e (211111)
%t partsabove[y_]:=Length[Select[Range[Length[y]],#<y[[#]]&]];
%t DeleteCases[Table[Length[Select[IntegerPartitions[n],partsabove[#]==k&]],{n,1,15},{k,0,n-1}],0,2]
%Y Row sums are A000041.
%Y Row lengths are A000194, reversed A003056.
%Y Column k = 1 is A002620, reversed A238875.
%Y Column k = 2 is A097701.
%Y The version for permutations is A008292, opposite A123125.
%Y The weak version is A115720/A115994, rank statistic A257990.
%Y The version for compositions is A352524, weak A352525.
%Y The version for reversed partitions is A353319.
%Y A000700 counts self-conjugate partitions, ranked by A088902.
%Y A001522 counts partitions with a fixed point, ranked by A352827 (unproved).
%Y A064428 counts partitions w/o a fixed point, ranked by A352826 (unproved).
%Y A238352 counts reversed partitions by fixed points, rank statistic A352822.
%Y Cf. A000701, A006918, A008290, A008930, A114088, A177510, A219282, A238874, A300788, A352522.
%K nonn,tabf
%O 1,5
%A _Gus Wiseman_, May 21 2022